1. The problem statement, all variables and given/known data Let R be a ring containing two or more elements such that for each nonzero a in R, there exists a unique b in R such that a=aba. Show that R contains no divisors of zero. 2. Relevant equations The ring axioms. 3. The attempt at a solution I assumed that R contains divisors of zero. So assume there exist nonzero a,b in R such that ab=0. Then there exist unique c,d in R such that a=aca and b=bdb. Consider the expression ca+bd. If ca+bd=0, then multiplying on the left by a gives a=0 and multiplying on the right by b gives b=0. So assume ca+bd is not equal to zero. Then there exists unique e in R such that ca+bd=(ca+bd)e(ca+bd). Multiplying on the left by a gives a=aebd+aeca=ae(ca+bd). Substituting this expression for a into a=aca gives a=aca=ae(ca+bd)ca. From the uniqueness of c, e(ca+bd)c=c. This is as far as I've gotten. I don't know if I have made any progress or am just going in circles.